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Axioms 2014, 3(4), 369-379; doi:10.3390/axioms3040369

A Simplified Algorithm for Inverting Higher Order Diffusion Tensors

1
Department of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, Eindhoven NL-5600 MB, The Netherlands
2
Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, Eindhoven NL-5600 MB, The Netherlands
*
Author to whom correspondence should be addressed.
Received: 20 April 2014 / Revised: 5 November 2014 / Accepted: 7 November 2014 / Published: 14 November 2014
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Abstract

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann–Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices. View Full-Text
Keywords: Riemann–Finsler geometry; biomedical image analysis; HARDI; Einstein contracted product Riemann–Finsler geometry; biomedical image analysis; HARDI; Einstein contracted product
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Astola, L.; Sepasian, N.; Haije, T.D.; Fuster, A.; Florack, L. A Simplified Algorithm for Inverting Higher Order Diffusion Tensors. Axioms 2014, 3, 369-379.

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